- To convert an improper fraction to a mixed number:
1. Divide the denominator into the numerator.
2. The integer part of the mixed number is the result of the division.
3. The fraction part of the mixed number is remainder over the denomi-
Example 1 Convert to a mixed number.
1. 13 ÷ 6 = 2 with a remainder of 1.
2. The integer part is 2.
3. The fraction part is .
- To convert a mixed number to an improper fraction:
Example 4 Multiply and reduce your answer to lowest terms.
1. 3 × 5 = 15 is the new numerator.
2. 10 × 2 = 20 is the new denominator.
4. Both the numerator and denominator have a factor of 5. Canceling this
out, we have that in lowest terms is
• Dividing Fractions
1. Find the reciprocal of the second fraction.
2. Multiply the first fraction by the reciprocal of the second fraction.
Example 5 Divide and reduce your answer to lowest terms.
1. The reciprocal of is
2. By multiplying and then canceling the common factor of 3, we have:
• Finding a Common Denominator
The common denominator of a set of fractions is a number which is divisible
by the denominators of all the fractions. The easiest way of finding a common
denominator is to multiply all the denominators together.
Example 6 Find a common denominator of
9 · 10 · 6 = 540 is a common denominator of the three fractions.
You may want to find the least common denominator, which is the smallest
number that is a common denominator of the fractions. This will require more
work to do, but, in a longer problem, it may make the rest of the problem
1. Completely factor all denominators.
2. For each number that occurs in the factorizations, find the maximum num-
ber of times that number occurs in the factorization of a single denomina-
3. The factorization of the least common denominator must contain each
number that occurs in the factorizations, and they must be included the
number of times determined in Step 2.
Example 7 Find the least common denominator of
1. The denominators factor as follows: 9 = 3 · 3, 10 = 2 · 5, and 6 = 2 · 3.
2. 2 and 5 occur at most once in any denominator, but 3 occurs twice in the
factorization of 9.
3. The factorization of the least common denominator must contain one 2,
one 5, and two threes: 2 · 5 · 3 · 3 = 90.
So, the least common denominator of
• Rewriting Fractions with a Common Denominator
1. Find a common denominator of the fractions.
2. For each fraction:
(a) Divide the common denominator by the denominator of the fraction.
(b) Multiply the numerator and the denominator by the result of the di-
vision. (Note that this does not change the value of the fraction since
it is really multiplication by one.)
Example 8 Rewrite
with a common denominator.
1. As seen in Example 7, the least common denominator is 90.